Camera with different properties
Afaik, you set the resolution to the "scene render properties", not "camera properties", and it makes sense, eg: if you animate a scene with two cameras, one landscape and the other portrait, and during the animation you switch the camera, how should the animation video be rendered...?

You can use a scene copy, though. Every scene has its own "scene render properties", and you can use Scene 1 render 1200 X 1800  and Scene 2 1800 X 1200. But you'll need to render them separately, I guess.